Quasimorphism

In mathematics, given a group $G$, a quasimorphism (or quasi-morphism) is a function $f:G\to\mathbb$ which is additive up to bounded error, i.e. there exists a constant $D\geq 0$ such that $, f(gh)-f(g)-f(h), \leq D$ for all $g, h\in G$. The least positive value of $D$ for which this inequality is satisfied is called the defect of $f$, written as $D(f)$. For a group $G$, quasimorphisms form a subspace of the function space $\mathbb^G$.

Examples

* Group homomorphisms and bounded functions from $G$ to $\mathbb$ are quasimorphisms. The sum of a group homomorphism and a bounded function is also a quasimorphism, and functions of this form are sometimes referred to as "trivial" quasimorphisms.Frigerio (2017), p. 12.
* Let $G=F_S$ be a free group over a set $S$. For a reduced word $w$ in $S$, we first define the big counting function $C_w:F_S\to \mathbb_$, which returns for $g\in G$ the number of copies of $w$ in the reduced representative of $g$. Similarly, we define the little counting function $c_w:F_S\to\mathbb_$, returning the maximum number of non-overlapping copies in the reduced representative of $g$. For example, $C_(aaaa)=3$ and $c_(aaaa)=2$. Then, a big counting quasimorphism (resp. little counting quasimorphism) is a function of the form $H_w(g)=C_w(g)-C_(g)$ (resp. $h_w(g)=c_w(g)-c_(g))$.
* The rotation number $\text:\text^+(S^1)\to\mathbb$ is a quasimorphism, where $\text^+(S^1)$ denotes the orientation-preserving homeomorphisms of the circle.

Homogeneous quasimorphisms

A quasimorphism is homogeneous if $f(g^n)=nf(g)$ for all $g\in G, n\in \mathbb$. It turns out the study of quasimorphisms can be reduced to the study of homogeneous quasimorphisms, as every quasimorphism $f:G\to\mathbb$ is a bounded distance away from a unique homogeneous quasimorphism $\overline:G\to\mathbb$, given by :
:$\overline(g)=\lim_\frac$.
A homogeneous quasimorphism $f:G\to\mathbb$ has the following properties:
* It is constant on conjugacy classes, i.e. $f(g^hg)=f(h)$ for all $g, h\in G$,
* If $G$ is abelian, then $f$ is a group homomorphism. The above remark implies that in this case all quasimorphisms are "trivial".

Integer-valued quasimorphisms

One can also define quasimorphisms similarly in the case of a function $f:G\to\mathbb$. In this case, the above discussion about homogeneous quasimorphisms does not hold anymore, as the limit $\lim_f(g^n)/n$ does not exist in $\mathbb$ in general.

For example, for $\alpha\in\mathbb$, the map $\mathbb\to\mathbb:n\mapsto\lfloor\alpha n\rfloor$ is a quasimorphism. There is a construction of the real numbers as a quotient of quasimorphisms $\mathbb\to\mathbb$ by an appropriate equivalence relation, see Construction of the reals numbers from integers (Eudoxus reals).

Notes

References

*
*{{Citation, last=Frigerio, first=Roberto, year= 2017, title=Bounded cohomology of discrete groups, series=Mathematical Surveys and Monographs, volume= 227 , publisher=American Mathematical Society, Providence, RI, isbn=978-1-4704-4146-3, doi=10.1090/surv/227, pages=12–15, arxiv=1610.08339, s2cid=53640921

Further reading

What is a Quasi-morphism?
by D. Kotschick

Mathematics
Additive functions